Extensional flow

An extensional flow stretches material along one or more directions while contracting it along others. A local ideal model of planar extension has velocity $\mathbf{u}=(\dot{ϵ}x,-\dot{ϵ}y,0)$: its rate-of-strain tensor is $\mathrm{diag}(\dot{ϵ},-\dot{ϵ},0)$ and it is an incompressible flow. Uniaxial extensional flow instead contracts both transverse directions. These geometries probe extensional viscosity and can stretch polymers.

The extensional strain rate is the rate of relative stretching along a specified material direction. If a short material line of length $\ell$ follows a principal stretching direction, $\dot{ϵ}=d\log \ell /dt$ is the corresponding eigenvalue of the rate-of-strain tensor. For the planar model in extensional flow, $\dot{ϵ}=\partial u_x/\partial x$; the two transverse rates are $-\dot{ϵ}$ and zero. For uniaxial extensional flow, each transverse rate is $-\dot{ϵ}/2$.